International Journal of Analysis and Applications ISSN 2291-8639 (Online)

Research Article

Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications

  • Izhar Ahmad1*
  • Krishna Kummari
  • S. Al-Homidan
  1. King Fahd University of Petroleum and Minerals

* Corresponding author

Abstract

In this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. Sufficient optimality conditions for a LU optimal solution are derived under generalized convexity assumptions. Moreover, appropriate duality results are discussed for a Mond-Weir type dual problem. In addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem.

How to Cite

I. Ahmad, K. Kummari, S. Al-Homidan, Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications, Int. J. Anal. Appl. 18 (2020), 784–798. https://doi.org/10.28924/2291-8639-18-2020-784

BibTeX
@article{Ahmad2020,
  author  = {Ahmad, Izhar and Kummari, Krishna and Al-Homidan, S.},
  title   = {Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications},
  journal = {Int. J. Anal. Appl.},
  volume  = {18},
  year    = {2020},
  pages   = {784--798},
  doi     = {10.28924/2291-8639-18-2020-784}
}

References

  • [1] J. M. Abadie, On the Kuhn-Tucker theorem, Nonlinear Programming, J. Abadie ed., John Wiley, New York, 1967, pp. 21-36.
  • [2] W. Achtziger and C. Kanzow, Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications, Math. Program. 114 (2008), 69-99.
  • [3] Y. An, G. Ye, D. Zhao and W. Liu, Hermite-Hadamard type inequalities for interval (h1, h2)-convex functions, Mathematics, 7 (2019), Art. ID 436.
  • [4] M. S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming. Theory and Algorithms. 2nd edition, John Wiley & Sons, Hoboken, 1993.
  • [5] M. P. Bendsøe and O. Sigmund, Topology Optimization-Theory, Methods and Applications, 2nd ed., Springer, Heidelberg, Germany, 2003.
  • [6] A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Math. Method. Oper. Res. 76 (2012), 273-288.
  • [7] R. W. Cottle, A theorem of Fritz John in mathematical programming, RAND Memorandum RM-3858-PR, RAND Corporation, 1963.
  • [8] M. Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space, SIAM J. Control, 7 (1969), 232-241.
  • [9] A. Khare and T. Nath, Enhanced Fritz John stationarity, new constraint qualifications and local error bound for mathematical programs with vanishing constraints, J. Math. Anal. Appl. 472 (2019), 1042-1077.
  • [10] Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, 1997.
  • [11] O. L. Mangasarian and S. Fromovitz, The Fritz John necessary optimality condition in the presence of equality and inequality constraints, J. Math. Anal. Appl. 17 (1967), 37-47.
  • [12] S. K. Mishra, V. Singh, V. Laha and R. N. Mohapatra, On constraint qualifications for multiobjective optimization problems with vanishing constraints, Optimization Methods, Theory and Applications, Springer Berlin Heidelberg (2015), 95-135.
  • [13] R. Osuna-G ´omez, B. Hern ´andez-Jim ´enez, Y. Chalco-Cano and G. Ruiz-Garz ´on, New efficiency conditions for multiobjective interval-valued programming problems, Inform. Sci. 420 (2017), 235-248.
  • [14] J. V. Outrata, M. Koˇcvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Nonconvex Optimization and its Applications, Kluwer, Dordrecht, 1998.
  • [15] D. W. Peterson, A review of constraint qualifications in finite-dimensional spaces, SIAM Rev. 15 (1973), 639-654.
  • [16] A. Sadeghi, M. Saraj and N. M. Amiri, Efficient solutions of interval programming problems with inexact parameters and second order cone constraints, Mathematics, 6 (2018), Art ID 270.
  • [17] D. Singh, B. A. Dar and A. Goyal, KKT optimality conditions for interval valued optimization problems, J. Nonlinear Anal. Optim. 5 (2014), 91-103.
  • [18] D. Singh, B. A. Dar and D. S. Kim, KKT optimality conditions in interval-valued multiobjective programming with generalized differentiable functions, European J. Oper. Res. 254 (2016), 29-39.
  • [19] M. Slater, Lagrange multipliers revisited: a contribution to nonlinear programming, Cowles Commission Discussion Paper, Mathematics, 403, 1950.
  • [20] I. M. Stancu-Minasian, Stochastic Programming with Multiple Objective Functions, D Reidei Publishing Company, Bordrecht, 1984.
  • [21] Y. Sun and L. Wang, Optimality conditions and duality in nondifferentiable interval- valued programming, J. Ind. Manage. Optim. 9 (2013), 131-142.
  • [22] Z. Wang and S. C. Fang, On constraint qualifications: motivation, design and inter-relations, J. Ind. Manage. Optim. 9 (2013), 983-1001.
  • [23] L. Wang, G. Yang, H. Xiao, Q. Sun, J. Ge, Interval optimization for structural dynamic responses of an artillery system under uncertainty, Eng. Optim. 52 (2020), 343-366.
  • [24] H. C. Wu, On interval valued nonlinear programming problems, J. Math. Anal. Appl. 338 (2008), 299-316.
  • [25] H. C. Wu, The Karush Kuhn Tuker optimality conditions in multiobjective programming problems with interval-valued objective functions, European J. Oper. Res. 196 (2009), 49-60.
  • [26] H. C. Wu, Solving the interval-valued optimization problems based on the concept of null set, J. Ind. Manage. Optim. 14 (3) (2018), 1157-1178.